Theorem eldprdi 19966

Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)

Hypotheses

Ref	Expression
eldprdi.0	⊢ 0 = (0g‘𝐺)
eldprdi.w	⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
eldprdi.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
eldprdi.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
eldprdi.3	⊢ (𝜑 → 𝐹 ∈ 𝑊)

Assertion

Ref	Expression
eldprdi	⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))

Distinct variable groups:   ℎ,𝐹   ℎ,𝑖,𝐺   ℎ,𝐼,𝑖   0 ,ℎ   𝑆,ℎ,𝑖

Allowed substitution hints:   𝜑(ℎ,𝑖)   𝐹(𝑖)   𝑊(ℎ,𝑖)   0 (𝑖)

Proof of Theorem eldprdi

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldprdi.1	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
2		eldprdi.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
3		eqid 2737	. . 3 ⊢ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)
4		oveq2 7378	. . . 4 ⊢ (𝑓 = 𝐹 → (𝐺 Σg 𝑓) = (𝐺 Σg 𝐹))
5	4	rspceeqv 3601	. . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)) → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
6	2, 3, 5	sylancl 587	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
7		eldprdi.2	. . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼)
8		eldprdi.0	. . . 4 ⊢ 0 = (0g‘𝐺)
9		eldprdi.w	. . . 4 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
10	8, 9	eldprd 19952	. . 3 ⊢ (dom 𝑆 = 𝐼 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
11	7, 10	syl 17	. 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
12	1, 6, 11	mpbir2and 714	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401   class class class wbr 5100  dom cdm 5634  ‘cfv 6502  (class class class)co 7370  Xcixp 8849   finSupp cfsupp 9278  0_gc0g 17373   Σ_g cgsu 17374   DProd cdprd 19941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-ixp 8850  df-dprd 19943

This theorem is referenced by:  dprdfsub  19969  dprdf11  19971  dprdsubg  19972  dprdub  19973  dpjidcl  20006